Computational Studies of Two-dimensional Crystals

Computational studies of two-dimensional crystals

Thesis presented by WENDELSILVAPAZ

for the degree of Doctor in Physics

DEPARTAMENTO DEFÍSICA DE LAMATERIACONDENSADA

FACULTAD DECIENCIAS

UNIVERSIDAD AUTÓNOMA DEMADRID

TESIS DOCTORAL

THESISSUPERVISOR: JUANJOSÉPALACIOS

MADRID, 2017

Estudio Computacional de Cristales Bidimensionales

Tesis presentada por WENDELSILVAPAZ

para optar al grado de doctor en Ciencias Físicas

DEPARTAMENTO DEFÍSICA DE LAMATERIACONDENSADA

FACULTAD DECIENCIAS

UNIVERSIDAD AUTÓNOMA DEMADRID

TESIS DOCTORAL

DIRECTOR DE TESIS: JUANJOSÉ PALACIOS

MADRID, 2017

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BSTRACT

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anoscale materials have become a field of high interest in research not only from a fundamental point of view, but also due to the potential of these materials in applied electronics, optoelectronics and spintronics. The present thesis is focused on the study of some of the most promising 2D materials. The research is conducted from an theoretical point of view. However, along the whole thesis we have been supported by experimental colaborations, which have allowed us to get a better comprehension of our results, enriching our understanding of the physics behind.

Semiconductors of atomically thick monolayers, which can be combined to create van der Waals heterostructures where monolayers of multiple 2D materials are stacked vertically layer- by-layer, can also be stitched together seamlessly in-plane to form lateral heterojunctions. Lat- eral interfacing of atomic monolayers has opened up unprecedented opportunities to engineer two-dimensional heteromaterials. Yet, little is known about the nature of these newly created interfaces. Here we turn our attention to, arguably, the most promising 2D crystal to date, a single layer of MoS₂. In this thesis, we present a theoretical study of the electrical contact between the two most common crystallographic phases of MoS₂monolayer crystals: 2H (semiconductor) and 1T (metallic).

The fabrication of van der Waals heterostructures, artificial materials assembled by individual stacking of 2D layers, is among the most promising directions in 2D materials research. Until now, the most widespread approach to stack 2D layers relies on deterministic placement methods, which are cumbersome and tend to suffer from poor control over the lattice orientations and the presence of unwanted interlayer adsorbates. Here, we present an extensive theoretical and experimental characterizations of franckeite which is a naturally occurring and air stable van der Waals heterostructure. As the bulk material is already composed of these alternating SnS₂ and PbS layers, the exfoliation process minmizes stacking missorientation and avoids interlayer adsorbates in the isolated nanosheets of franckeite. Hence, franckeite can be considered as a naturally occurring vdW heterostructure analog of its synthetic cousin.

One of the main problems that the scientific community is facing nowadays in nanostructured electronics is the heat dissipation issue that usually leads to device malfunction. Therefore, great efforts are being dedicated to find new materials that could circunvent this problem and to understand the thermal mechanisms working in nanoscale electronics. In this thesis, we investigate the electrical breakdown of TiS₃ nanoribbon-based field-effect transistors (FETs) and the thermal mechanisms that lead to the devices breakdown. Furthermore, these results are compared with thermogravimetric analysis of bulk TiS₃ degradation, as well as with density functional theory and Kinetic Monte Carlo simulations of surface oxidation and the subsequent desorption of sulphur atoms that lead to the creation of defects and could explain the FETs malfunction.

The recent isolation of antimonene, a novel two-dimensional material, pushes even further

BL) antimonene shows metallic characteristic. In particular, few-layer (> 6 BL) presented the gapless configuration of the surface bands give rise to a single Dirac cone (double degenerate), which is the signature of nontrivial topological order. H₂O molecules on the surface broke the degeneracy of the Dirac cone giving rise to two Dirac cones at theΓ point separated by 60 meV.

The experimental electrical properties reported herein are in good agreement with theoretical calculations still ongoing, pointing to a conduction governed by topologically protected surface states in few-layer antimonene.

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ESUMEN

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os materiales a nanoescala se han convertido en un campo de gran interés en la inves- tigación no sólo desde un punto de vista fundamental, sino también debido al potencial de estos materiales en la electrónica aplicada, la optoelectrónica y la espintrónica. La presente tesis se centra en el estudio de algunos de los materiales 2D más prometedores. La investigación se lleva a cabo desde una perspectiva teórica. Sin embargo, a lo largo de toda la tesis hemos sido apoyados por colaboraciones experimentales, que nos han permitido comprender mejor de nuestros resultados, enriqueciendo nuestra comprensión de la física que hay detrás.

Los semiconductores de monocapas atómicamente gruesas, que se pueden combinar para crear heterostructuras de van der Waals, donde las monocapas de múltiples materiales 2D se apilan verticalmente capa por capa, también pueden ser cosidas juntas de forma transparente en el plano para formar heterojunciones laterales. La interfaz lateral de monocapas atómicas ha abierto oportunidades sin precedentes para diseñar heteromateriales bidimensionales. Sin embargo, poco se sabe sobre la naturaleza de estas interfaces recién creadas. Aquí dirigimos nuestra atención a, posiblemente, el más prometedor cristal 2D hasta la fecha, una sola capa de MoS₂. En esta tesis, presentamos un estudio teórico del contacto eléctrico creado entre las dos fases cristalográficas más comunes de los cristales monocapa de MoS₂: 2H (semiconductor) y 1T (metálico).

La fabricación de heterostructuras de van der Waals, materiales artificiales ensamblados por apilamiento individual de capas 2D, está entre las áreas más prometedoras en la investigación de materiales 2D. Hasta ahora, el enfoque más difundido del apilamiento de las capas 2D se basa en métodos de colocación deterministas, que son engorrosos y tienden a sufrir de un control deficiente sobre las orientaciones de la red y la presencia de adsorbidos de intercapa indeseados.

Aquí, presentamos extensas caracterizaciones teóricas y experimentales de la Franckeíta que es una heterostructura de van der Waals natural y estable al aire. Como el material a granel ya está compuesto por estas capas alternas de SnS₂y PbS, el proceso de exfoliación reduce al mínimo la desorientación del apilamiento y evita adsorbatos intercapa y en las monocapas aisladas de la Franckeíta. Por lo tanto, la Franckeíta puede ser considerada como una heteroestructura de van der Waals de origen natural análoga a su primo sintético.

Uno de los principales problemas que enfrenta la comunidad científica hoy en día en elec- trónica nanoestructurada es el problema de la disipación de calor que suele conducir a un mal funcionamiento del dispositivo. Por lo tanto, se están dedicando grandes esfuerzos a encontrar nuevos materiales que puedan evitar este problema y comprender los mecanismos térmicos que funcionan en la electrónica a nanoescala. En esta tesis, investigamos la descomposición eléctrica de los transistores de efecto de campo (FET) basados en nanocintas de TiS₃ y los mecanismos térmicos que conducen a la avería de los dispositivos. Además, estos resultados se comparan con el análisis termogravimétrico de la estructura granel, así como con los resultados obtenidos a través de la teoría del funcional de la densidad y las simulaciones de Monte Carlo cinético de la

El reciente aislamiento del Antimoneno, un nuevo material bidimensional, empuja aún más el interés por este material que ya había sido previsto teóricamente. Aquí, presentamos resultados preliminares sobre las propiedades electrónicas y eléctricas del antimoneno de pocas capas. El Antimoneno de pocas capas (> 2 BL) tiene una característica metálica. En particular, el caso (> 6 BL) presentó una configuración sin ranuras de las bandas de superficie dando lugar a un solo cono de Dirac (Doble degenerado), que es la firma del orden topológico no trivial. La presencia de moléculas de H₂O en la superficie rompió la degeneración del cono de Dirac dando lugar a dos conos Dirac en el puntoΓ separados por 60 meV. Las propiedades eléctricas experimentales presentadas hasta ahora están en buen acuerdo con los cálculos teóricos aún en curso, apuntando a una conducción regida por estados de superficie topológicamente protegidos en el Antimoneno de pocas capas.

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CKNOWLEDGEMENTS

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he most important influence on the successful completion of this thesis was my advisor, Prof. Juan José Palacios. He has had greatest professional influence on my development as a scientist. He is an amazing scientist and mentor. He pushed me to develop my weaknesses and exploit my strengths. His courage to tackle new and difficult problems and his patience to withstand the many failures that accompany such risks is admirable.

Most importantly, I would like to thank my family for all the support and encouragement during my studies. I want to thank my parents. It is their love and support through the years that brought me to Madrid for proving that there is life and meaning also outside of science.

I would like to thank all collaborators of this work: Aday Molina-mendonza; Pablo Ares and Sahar Pakdel who have been working closely and also for the other collaborators, without her help this study could not be possible. I would like to express my gratitude to the colleagues who helped me a lot during the course of this study: María Soriano, Mohammed, Daniel Bejarano, Manrico and Carlos Salgado.

I am also truly grateful to the Autonomous University of Madrid and like to thank the great support staff and the secretaries that take care of the paperwork and negotiate the grand bureau- cracies of the academic world. I would like to the computer resources and assistance provided by the Centro de Computación Científica of the Universidad Autónoma de Madrid and the RES.

I am also thankful to the support from CAPES: Science Without Borders Program, to the Ministry of Education in Brazil, for my scholarship.

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ONTENTS

Page

List of Tables xi

List of Figures xiii

1 Introduction 1

1.1 Electrical contacts to 2D materials . . . 3

1.1.1 Lateral heterostructures . . . 5

1.2 van der Waals heterostructures . . . 6

1.3 New promising 2D materials . . . 7

1.3.1 Germanene . . . 7

1.3.2 Antimonene . . . 8

2 Theoretical Background 11 2.1 Introduction . . . 11

2.2 Tight-binding model. . . 13

2.2.1 Electrons is solids . . . 13

2.3 Linear Combination of atomic orbital (LCAO) . . . 13

2.3.1 Crystal and atomic hamiltonians . . . 13

2.3.2 The atomic wavefunctions . . . 14

2.3.3 Bloch’s theorem. . . 15

2.4 The Slater-Koster approximation . . . 16

2.5 Electronic transport. . . 19

2.5.1 Landauer formalism . . . 19

2.5.2 Green’s functions and partitioning technique . . . 23

3 A theoretical study of the electrical contact between metallic and semicon- ducting phases in monolayer MoS₂ 25 3.1 Introduction . . . 25

3.2 Methods . . . 28

3.3 Results and discussion . . . 29

3.4 Conclusion . . . 37

4 Franckeite: a naturally occurring stacked van der Waals heterostructure with a narrow bandgap 39 4.1 Introduction . . . 39

4.2 Experimental setup and characterization . . . 40

4.2.1 Crystal structure . . . 40

4.2.2 X-ray diffraction characterization . . . 40

4.2.3 TEM and XPS of mechanically exfoliates flakes . . . 42

4.2.4 Absorption spectroscopy. . . 44

4.2.5 Scanning tunneling spectroscopy . . . 46

4.3 Electronic structure of Franckeite . . . 47

4.3.1 (PbSnS) and SnS₂-based franckeite . . . 47

4.3.2 Semiconductor heterostructure . . . 48

4.3.3 Thickness dependent band structure . . . 52

4.3.4 Sb-doped franckeite . . . 52

4.3.5 Band structure with SOC . . . 54

4.4 Conclusion . . . 56

5 High Current Density Electrical Breakdown of TiS3Nanoribbon-Based Field- Effect Transistors 59 5.1 Introduction . . . 59

5.2 Crystal structure and synthesis. . . 60

5.3 TiS₃-based field-effect transistors . . . 61

5.3.1 Electrical breakdown of FETs . . . 63

5.4 Density functional theory calculations of activation energies . . . 67

5.5 Object kinetic Monte Carlo algorithm . . . 69

5.6 Conclusion . . . 71

6 Antimonene: a new bidimensional material 73 6.1 Introduction . . . 73

6.2 Crystal structure . . . 73

6.3 Allotropes and structural stability . . . 74

6.4 Monolayer, bilayer and trilayer Antimonene . . . 75

6.5 Few-layer Antimonene . . . 75

6.6 Effect of H₂O molecules on the 7BL Sb . . . 77

6.7 Electrical characterization of few-layer antimonene. . . 79

6.8 Conclusion . . . 82

7 General conclusions 83

TABLE OF CONTENTS

8 Conclusiones generales 87

A Appendix A 91

A.1 Density Functional Theory (DFT) . . . 91

A.1.1 The Hohenberg-Kohn theorems . . . 92

A.1.2 The Kohn-Sham density-functional . . . 93

A.1.3 Exchange-correlation functionals . . . 93

A.1.4 HSE Hybrid functional . . . 95

B Appendix B 97 B.1 Thermopower. . . 97

B.2 Raman spectroscopy . . . 99

B.3 Liquid phase exfoliation . . . 100

B.4 Franckeite-based nanodevices. . . 102

B.4.1 Electronic characterization . . . 102

B.4.2 Optoelectronic characterization . . . 103

B.5 MoS₂-franckeite p − n junction . . . 105

C Appendix C 109 C.1 Thermogavimetric analysis and mass spectroscopy . . . 109

References 113

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ABLES

TABLE Page

1.1 Summary of properties of different 2D materials relevant for nanodevices applications. 2 2.1 Hopping integrals for s, p, and d orbitals. The hopping integrals not given in this

table can be found by cyclically permuting the directos cousines. . . 20 3.1 Tight-binding parameters for single-layer MoS₂. All the hoppings terms V_αand crystal

fields∆αare in unit of eV. . . 29 4.1 Fundamental band gaps (in eV) of H layer, Q layer, franckeite crystal and franckeite

with 50% Sn and 50% Sb in the Q layer and 100% Sb in the H layer.. . . 55 5.1 Binding energy for the oxygen impurities for different concentrations. . . 68

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IGURES

FIGURE Page

1.1 Schematic of a typical 2D semiconductor FET. The figure states key issues affecting 2D device performance related to channel materials, metal contact, 2D dielectric interfaces, and short channel effects. (extracted from [43]). . . 3 1.2 Current strategies to improve electrical contacts in 2D semiconductor FETs. a) Play

with types of contact metals and processing recipe. b) Doping of contact areas. c) 1T-2H phase-engineered in-plane heterojunctions. d) Graphene top contact. e) Graphene edge contact. f) Inserting a tunnel layer between 2D semiconductors and contact electrodes.

(extracted from [43]). . . 4 1.3 Schematic diagrams of a device with a 1T’/2H phase homojunction. (B) AFM image

of a device with the 1T’/2H phase homojunction in MoTe2. (C) Raman mapping images of 1T’ (A_g) and 2H(E_2g) vibrational modes in the device channel in (B). (D) Source-drain current I_SD characteristics for gate voltage V_G ranging from -60 V to 60 V. (E) Arrhenius plots of the conductance. (F) Field-effectmobility as a function of temperature (extracted from [51]). . . 5 1.4 Many functional van der Waals heterostructures can be created. What started with

mechanically assembled stacks (top) has now evolved to large-scale growth by CVD or physical epitaxy (bottom) (extracted from [52]). . . 7 2.1 1D TB scheme . . . 14 2.2 Sketch of transport problem for the case of one-dimensional nanowires as electrodes.

The system is divided into three parts: L, D, and R. Extracted from [80] . . . 21 3.1 High-resolution transmission electron microscope image of an atomically thin phase

boundary (indicated by the arrows) between the 1T and 2H phases in a monolayered MoS₂ nanosheet. Scale bar, 5 nm. (Extracted from [110]). . . 26 3.2 Resistance versus 2H channel lengths for Au deposited directly on the 2H phase (a,b)

and on the 1T phase (c,d). (Extracted from [110]) . . . 27 3.3 (color online). Sketch of the model used to calculate the electronic transport properties

of 2H/1T monolayer with armchair and zigzag interfaces. . . 29

3.4 (color online). Atomic structure of semiconducting phase (2H) and a metallic phase (1T). The 2H phase shows hexagon lattice and an atomic stacking sequence of AB. The 1T phase shows the atomic stacking sequence of ABC. . . 30 3.5 (color online). (a) Band structure of a 1T and (b) 2H-zigzag nanoribbon (3.1 nm); (c) 1T

and (d) 2H-armchair nanorribon (2.5 nm) of MoS₂. Blue lines correspond to the edges states while dashed lines represents empty bands. The CNL is at zero. . . 31 3.6 (color online). Schematic energy diagram at the 1T/2H interface. On the left the Mott-

Schottky band-alignment is represented. On the right, the band-bending due to the charge dipole and their expected decay away from the interface is shown for both crystallographic orientations of the interface. . . 32 3.7 Partial density of states (pDOS) for the 2H/1T (a) armchair interface and (b) zigzag

interface. (Red lines) Mo atom; (green lines) S atoms; (black lines) Mo + S atoms. The indices 1-5 represent the unit cell of the 2H phase from the interface. The Fermi level is to zero and the dashed black lines represent the CNL.. . . 33 3.8 (color online). Isosurface (0.015 e/Å⁻³) of the DOS at the Fermi energy for zigzag (a)

and armchair (b) interfaces showing the larger extension into the bulk of the edge states for the zigzag case.. . . 34 3.9 (color online). Trasmission curves for 2H-MoS₂nanoribbons with transport direction

perpendicular to armchair direction (a), the same for the 2H/1T armchair interface (b), the same for 2H-MoS₂ nanoribbons with transport direction perpendicular to zigzag direction, and similarly for the 2H/1T zigzag interface. . . 35 3.10 (color online). Transmission curves for holes (a,c) and electrons (b,d) close to conduction

band maximum (CBM) and valence band maximum (VBM), respectively. The dashed black lines represent the transmission curves for standard onsite terms as shown in Table 3.1 and the dashed green lines represent the transmission curves after adding the DFT dipole potential to the on-site energies. . . 35 3.11 (color online). Contact resistance for holes with standard onsite energies (a,c) (black

lines) and changed onsite energies (b,d) (green lines)._δE indicates the disorder. . . . 36 3.12 (color online). Contact resistance for electrons with standard onsite energies (a,c) (blue

lines) and changed onsite energies (b,d) (red lines)._δE indicates the disorder parameter. 37 4.1 (color online). (left) 2D and (right) 3D sketches of the crystal structure of franckeite:

the Q layer includes MX compounds, where M = Pb²⁺, Sn²⁺or Sb³⁺and X = S, while the H layer includes MX₂ compounds, where M = Sn⁴⁺(M can also be Fe²⁺or Sb⁴⁺

replacing Sn⁴⁺) and X = S. . . 41 4.2 Photograph of the franckeite mineral. (b) Zoom-in of the mineral shown in (a). . . 41 4.3 (color online). X-ray diffraction pattern measured in powder franckeite. . . 41

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4.4 (a) HRTEM micrograph of a franckeite sheet exhibiting the characteristic fringes of franckeite due to the corrugation induced by the misfit between Q and H layers. The scale bar is 40 nm. (b) Representative atomic scale HRTEM of an ultrathin franckeite layer. The scale bar is 5 nm. (c) SAED diagram consistent with a misfit layer compound made of PbS and SnS₂ layers: Q (purple) and H sublattices (light blue) lead to the most intense reflections on which superlattice rows of weak intensity are centred. The diagram has been indexed using tetragonal and orthohexagonal vectors for the Q and H phases respectively, according to the orientation and nomenclature defined in ref.

23. (d) Sb 3d3/2 and Sn 3d3/2 XPS spectrum acquired with photon energy h_ν= 600 eV. Inset: LEEM image (the field of view is 50 mm and the electron energy is 0.12 eV), the red square indicates the region of integration where the XPS spectra has been acquired. (e) S 2p1/2 and 2p3/2 and Pb 4 f 5/2 and 4 f 7/2 XPS spectrum acquired with photon energy h_ν= 230 eV. Inset: XPEEM image at Pb 4 f 7/2 core level (the field of view is 50 mm and the photon energy is 230 eV). The strong background in the XPS spectra is due to the tail of secondary electrons cascade. . . 43 4.5 XPS spectra of Fe 2p region (left), coincident with Sn 3p3/2 core level and Ag 3d

region (right).. . . 44 4.6 (a) SEM image of several ultrathin franckeite layers deposited on a TEM grid. (b),

Zoom in on the white squared area marked in a, showing layers of various thicknesses.

Conditions: 15 kV, 11 pA, working distance 6.5 mm. c, Zoom in on the image shown in (b). . . 45 4.7 UV-Vis-NIR spectrum of a thin film of franckeite colloidal suspension deposited on a

glass slide; the sample originates from the liquid-phase exfoliation of a 100 mg·mL⁻¹ franckeite powder dispersion. Inset: zoom of the region indicated by a dashed red line that highlights the absorption peak around 2900 nm.. . . 45 4.8 (a) Absorption spectra of the sample shown in Figure 4.7 and three replicates per-

formed with other three different glass slides. Inset: zoom in the 2500 nm-3300 nm region. (b) Absorption spectra normalized at 350 nm for the four samples presented in (a). Inset: zoom in the 2500 nm-3300 nm region. . . 46 4.9 Optical microscopy photograph of a piece of franckeite. The material presents large

area flat terraces after mechanical exfoliation. (b) STM topographic image of bulk franckeite. (c) STS characterization of franckeite, here we show the average curve obtained from a set of 200 current-voltage curves, which clearly shows a semiconduct- ing p-doped since the zero bias voltage (coinciding with the Fermi level) is closer to the valence band than to the conduction band. Inset: STS current-voltage curves in logarithmic scale, the valence and conduction bands values (-0.25 eV and 0.35 eV, respectively) are highlighted with arrows, yielding an electronic bandgap of ∼ 0.6 eV (∼ 0.7 eV after correction of the thermal broadening).. . . 47

4.10 (a) GGA band structure of the H and (b) Q layers. . . 48 4.11 Relative positions of valence/conduction band, gap energy and Fermi level for two

generics semiconductors. . . 49 4.12 Energy band diagram of type I, type II, and type III heterojunction interfaces. . . 50 4.13 Band alignment based on the work function (W) of Q layer (semimetal) and electron

affinity_χof H layer (semiconductor). (q is the electron charge).. . . 50 4.14 (a) Calculated band structure for the Q layer indicating the valence band maximum

(VBM) and conduction band minimum (CBM) (red lines). (b) Calculated band structure for the H layer indicating the valence band maximum (VBM) and conduction band minimum (CBM) (blues lines). (c) Calculated band structure for the franckeite crystal that presents a bandgap of ∼ 0.5 eV. The valence band is given by the H layer (red line), while the conduction band is given by the Q layer (blue line), suggesting that franckeite is a type-II heterostructure.. . . 51 4.15 (a) Crystal structure of franckeite indicating the H and Q layers. (b) Bloch states in

franckeite in which the valence (red) and conduction (blue) bands are represented.. . 52 4.16 The evolution of the band structure of Franckeite with 4 increased thickness for (a) n

= 1. (b), n = 2. (c), n = 3 and (d), bulk. . . 53 4.17 (a) Calculated band structure of franckeite without Sb. (b) Calculated band structure

of franckeite when 50% of the Sn atoms have been replaced by Sb atoms in the H layer. (c) Calculated band structure of franckeite when 100% of the Sn atoms have been replaced by Sb atoms in the H layer. . . 53 4.18 (a) Calculated band structure of franckeite with 50% Sn and 50% Sb in the Q layer

and 0% Sb in the H layer. (b) Calculated band structure of franckeite with 50% Sn and 50% Sb in the Q layer and 50% Sn and 50% Sb in the H layer. (c) Calculated band structure of franckeite with 50% Sn and 50% Sb in the Q layer and 100% Sb in the H layer. . . 54 4.19 (a) SnS₂(H layer). (b) PbSnS (Q layer) and (c) bulk franckeite from DFT/PBE with

spin-orbit coupling. . . 55 4.20 (a) Calculated band structure of franckeite with 50% Sn and 50% Sb in the Q layer

and 100% Sb in the H layer without considering SOC. (b) The same but considering SOC. . . 56 5.1 a) Artistic representation of the TiS₃unit cell where the gray spheres represent the

Ti atoms and the yellow spheres represent the S atoms. b) Artistic representation of the layered crystal structure of TiS₃. The unit cell is indicated by solid black lines. c) Photograph of TiS₃ inside the ampoule used to grow the material. Inset: magnified photograph of TiS₃ ribbons. d) SEM image of the TiS₃powder showing the nanoribbon morphology of the material. e) Raman spectra acquired in the TiS₃powder . . . 60

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5.2 a) SEM image of TiS₃ FET fabricated from one TiS₃nanoribbon. b) Higher resolution SEM image of one of the TiS₃ FETs shown in (a). c) Current-voltage characteristics of a TiS₃ nanoribbon device for different back-gate voltages. d) Transfer curves of a TiS₃ nanoribbon FET for different drain-source voltages. . . 62 5.3 (a) AFM image of the TiS₃nanoribbon device used to estimate the contact resistance.

(b) Plot of the two terminal channel resistance (V_b= 500 mV, V_g= 0V) as a function of channel length.. . . 62 5.4 a) Current-voltage characteristics during the electrical breakdown process of two

TiS₃ devices. The green circles highlight the current and voltage values just before breakdown. AFM topographic images of one device b) before and c) after electrical breakdown, including the AFM line profile that indicates the thickness. . . 63 5.5 Maximum current density at breakdown for different nanomaterials: multilayer MoS₂

[171], multilayer graphene [172], monolayer graphene [173], Cu [174], TiO₂[175], Si [176] and GaN [177] and TiS₃ (present work). . . 64 5.6 Current-voltage curve of a TiS₃nanoribbon-based device measured in vacuum (P <

10⁻⁵mbar). The electrical breakdown occurs at a current density of 9.5·105 A/cm², in the same range as the devices measured in air. . . 65 5.7 Current density at breakdown voltage versus resistivity in logarithmic scale for all

the measured devices (blue dots). The red line represents a linear fit which yields a slope of -0.78. . . 65 5.8 Calculated temperature distribution at breakdown along four TiS₃nanoribbons-based

devices using the 1D heat equation. The zero position (dashed gray line) corresponds to the center of the nanoribbon, while the -250 and 250 nm correspond to both source and drain electrodes edges. The highest temperature is reached in the center of the device and oscillates between 350 and 450 °C. . . 66 5.9 Artistic drawing of the different process considered in the DFT calculations. (a) S

atoms desorption from the surface. (b) SO pairs desorption from the surface, creating a mono-vacancy. (c) SO pairs desorption from the surface, creating a di-vacancy. . . . 68 5.10 Calculated evolution of the relative desorption of SO (∆m) atoms on the surface layer

of TiS₃ as a function of the temperature for different time rates: 1 min (light blue), 5 min (medium-dark blue), and 10 min (dark blue). . . 71 6.1 Relevant views and parameters of antimony atomic lattice (Extracted from [16]).. . . 74 6.2 Structural configurations of antimonene allotropes: (a)_α-Sb, (b)_β-Sb, (c)_γ-Sb and (d)

δ-Sb.( Extracted from [190]). . . 75 6.3 Band structure with one to three bilayer thickness. The Fermi level is set to zero. . . 76 6.4 Band structure with five to seven bilayer thickness. The Fermi level is set to zero. . . 76 6.5 Relaxed structure of 7BL Sb with 3 layers of H₂O on the surface. . . 78

6.6 Band structure of 7BL antimonene. Solid (dashed) lines correspond to the layer in the top (bottom). Blue (red) correspond to spin down (up). . . 78 6.7 Setup for the electrical characterization of FL-Sb flakes. a) AFM topographic image of

the whole area under study. This image is a collage of AFM images of smaller areas following the Au path, hence the absence of any feature far from it. A diagram of the electrical circuit for one of the flakes has been included for the sake of clarity. On the bottom left corner a micrometer size gold electrode created by thermal evaporation assisted by stencil mask can be seen. b) Details of the studied FL-Sb flakes. Flakes are designated with the numbers shown in the bottom right corners: 1, 2 and 3 from left to right. . . 79 6.8 Electrical characterization of FL-Sb flakes. a) AFM topographic image of flake 1. Spots

where IV curves at different tip-electrode distances are marked with different symbols.

b) Representative IV curves at different tip-electrode distances for the lowest terrace in flake 1. c) Resistance vs. Length plots obtained from IV curves in the different terraces in flake 1 as pointed by the symbols in a). Dashed lines are linear fits for data from each terrace. No significant differences in the slopes are found within error (contact resistance was subtracted in this plot for clarity). d) Resistance vs. Length plots from similar IV curves as in b) obtained in the different flakes using different AFM tips. Red: flake 1, blue: flake 2, gray: flake 3. Symbols are experimental values and lines are linear fits to these experimental data. Solid symbols with solid fit lines correspond to data acquired with one tip whereas empty symbols with dashed fit lines correspond to data acquired with another tip. . . 80 6.9 Partial density of states (PDOS) of 3 to 15BL Sb thickness. . . 81

A.1 LDA scheme . . . 94

B.1 Equivalent thermal circuit of the setup for measuring the thermopower. The sample is kept at ambient temperature T_Cold while the tip is heated to a temperature of T_{H ot}= TCold+∆· S is the thermopower of franckeite and Slead is the thermopower of the tip connecting lead. V_{BI AS}is the bias voltage applied at the sample. . . 97 B.2 Equivalent thermal circuit of the setup for measuring the thermopower. The sample

is kept at ambient temperature T_Cold while the tip is heated to a temperature of T_{H ot}= TCold+∆· S is the thermopower of franckeite and Slead is the thermopower of the tip connecting lead. V_{BI AS}is the bias voltage applied at the sample. . . 98 B.3 Raman spectra of franckeite raw powder (blue line) and liquid-phase (LP)-exfoliated

franckeite obtained from the sonication of a 100 mg·mL⁻¹powder dispersion in NMP (pink line).. . . 99

LIST OFFIGURES

B.4 Franckeite samples. Left: bulk mineral; middle: powder material obtained after grind- ing of raw chips; right: suspension of exfoliated material prepared by sonication of a 100 mg·mL⁻¹powder dispersion in NMP. . . 100 B.5 (a) AFM micrograph of the sample drop-casted and dried over a freshly exfoliated

mica substrate. (b) Statistical analysis of the raw height data. (c) and (d) TEM images of repre-sentative franckeite nanosheets prepared by exfoliation of a 100 mg·mL⁻¹ powder dispersion in NMP . . . 101 B.6 (a) AFM topographic characterization of franckeite nanosheets obtained from the exfo-

liation of a 1 mg·mL⁻¹powder dispersion in isopropanol/water 1/4 (v/v). (b) Statistical analysis of the AFM raw height data. The inserted numbers indicate the corresponding number of layers (unit cell, H + Q layer, 1.7 nm in thickness) from 4 layers (4L) up to

∼ 13 layers (13L). (c) and (d) TEM images of of franckeite nanosheets obtained from the exfoliation of a 1 mg·mL⁻¹powder dispersion in isopropanol/water 1/4 (v/v). . . . 101 B.7 (a) AFM topographic image of a franckeite-based device. The scan profile yields a

thickness ranging from 7 nm to 13.5 nm. (b) Optical microscopy image of the device shown in (a). . . 102 B.8 Current as a function of the applied back-gate voltage in dark conditions for the device

shown in Figure B.7 (V_ds= 150 mV). The gate-dependence shows a p-type doping, hole conduction. The first measurement (blue line) was repeated after 41 days (pink line), showing a drop of 5%, yielding a good stability of the device. Inset: current-voltage curve with an applied back-gate voltage of -40 V. . . 103 B.9 (a) Current as a function of the applied back-gate voltage (V_ds= 150mV ) for the

device shown in Figure B.7 in dark conditions and upon illumination with a 640 nm wavelength laser with different powers. (b) Responsivity of the device shown in (a) upon illumination with a 640 nm wavelength laser as a function of the laser effective power with an applied back-gate voltage of V_g = -30 V and V_ds= 150 mV. (c) Current as a function of the applied back-gate voltage (V_ds = 150 mV) for the same device upon ilumination with lasers of different wavelengths at the same intensity (P_d= 6.3 mW· cm⁻²). There is photocurrent generation even at wavelengths as large as 940 nm.

(d) Photocurrent as a function of the laser wavelengths with the same light intensity for back gate voltages of -20 V and +20V and V_ds= 150 mV. . . 104 B.10 (a) AFM topographic image of a franckeite-based device. The scan profile yields a

thickness ranging from 7 nm to 13.5 nm. (b) Optical microscopy image of the device shown in (a). (c) Artistic representation of the p-n junction shown in (a). . . 105

B.11 (a) Diode-like current-voltage (I_ds− Vg) curve of the p − n junction in dark conditions for different applied back-gate voltages. Inset: gate trace extracted from the I_ds− Vg

at V_ds= 750 mV. The p − n junction switches on at an applied back-gate voltage of 0 V. (b) Diode-like current-voltage (I_ds− Vg) curve of the p − n junction at an applied back-gate voltage of V_g= 40 V in dark conditions and upon illumination with laser of 940 nm and 885 nm wavelength, both with a power of 140_µW. The inset highlights the region around V_ds= 0 V and I_ds= 0 V to show the short-circuit current (I_sc) and open circuit voltage (V_oc) values, obtaining I_sc= -27 pA and V_oc= 55 mV at 940 nm, and I_sc= -51 pA and V_oc= 77 mV at 885 nm. . . 106 B.12 P − n junction current-voltage characteristics for different back-voltages ranging from

-40 V to 40 V upon illumination with (a) 940 nm wavelength and (b) 885 nm wavelength.107 B.13 P − n junction electrical power harvested in the device, calculated as Pel=| Ids|· Vds

upon illumination with a laser spot of (a) 940 nm wavelength with a P_el,max∼ 0.5 pW and (b) 885 nm wavelength with a P_el,max ∼ 1.2 pW. . . 107 C.1 Ion current signal related to molecular oxygen (m/q = 32) during thermal decompo-

sition of TiS₃. An increase of oxygen consumption from T = 350 °C up to 550 °C is observed (O₂- ion current signal decreases) which is related to the reaction between oxygen and TiS₃. . . 110 C.2 a) Thermogravimetric analysis of bulk TiS₃ in oxygen atmosphere. Inset: Thermo-

gravimetric analysis of bulk TiS₃in argon around the first event. b) XRD pattern of TiS₃as-synthetized (blue line) and after thermal treatment (red line). The blue circles correspond to TiS₃ phase, the red circles correspond to TiS₂ phase, * corresponds to TiO₂ anatase, and the black circles correspond to TiO₂rutile. . . 110 C.3 (a) Thermogravimetric curves obtained at different heating rates of TiS₃ under argon

atmosphere. (b) Kissinger plots and calculated activation energies of the two events.. 111

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Part of the contents of this work can be found in the following publications:

Molina-Mendonza, A. J., Giovanelli, E., Paz, Wendel S. et. al., Franckeite as a naturally ocorring van der Waals heterostructure, Nature Communications, 8, 14409 (2017)

Molina-Mendonza, A. J., Island, J. O., Paz, Wendel S. et. al., High Current Density Electrical Breakdown of TiS₃Nanoribbon-Based Field-Effect Transistors, Advanced Functional Materials, 27(13), 1605647 (2017)

Paz, Wendel S. and Palacios, J. J., A theoretical study of the electrical contact between metallic and semiconducting phases in monolayer MoS₂, 2D Materials, 4, 015014 (2016)

Scopel, W. L., Paz, Wendel S. and Freitas, Jair C. C., Interaction between single vacancies in graphene sheet: An ab initio calculation, Solid State Communications, 240, 5-9 (2016)

Freitas, J. C. C., Scopel, W. L., Paz, Wendel S. et. al., Determination of the hyperfine magnetic field in magnetic carbon-based materials: DFT calculations and NMR experiments, Scientific Reports, 5, 14761 (2015)

Preprints

Zhixiang, S., Maldonado, A., Paz, Wendel S., Natalya, Y. Shitsevalova, V. B, Filippov, D. S., Inosov, S, Andreas, P. S., Palacios, J. J. and Peter Wahl, Local Dopping in SmB₆ (2017). (sent to npj Quantum Materials)

C

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1

I

NTRODUCTION

I

n 2004, the isolation of graphene, one-atom-thin layer of hexagonally arranged carbon atoms, on an insulating surface [1] gave birth to a new era of atomically thin materials in solid-state electronics, which ultimately led to the search for additional two-dimensional materials[2–6]. These materials have a combination of properties that could make them key enablers for many applications, generating new products that cannot (or may be difficult to) be obtained with current technologies or materials [7]. Moreover, most of the 2D semiconductor families studied so far have also shown interesting phenomena, some of them observed for the first time in these 2D systems [8–11].

One group of such materials is the transition metal dichalcogenides (TMDC’s). Unlike graphene, which consists of an individual atom thick layer, TMDC’s follow a MX₂ structure, in which M is a transition metal (Mo, W, Re, Nb, etc) and X is a chalcogen atom (S, Se, or Te). De- pending on the combination of M and X, the resulting material can vary from semiconducting to metallic and even superconducting [12,13]. Other semiconducting 2D materials such as silicene [14], germanene [15] and antimonene [16] are also attracting growing interest. The electronic properties of silicene and germanene (the Si and Ge equivalent of graphene) have also been studied theoretically [17] both materials being predicted to be gapless semiconductors with linear energy dispersion relations near the K points, like graphene.

The broad family of layered materials is only so far barely explored and there still exist many members that remain to be investigated [18]. The group IV-V transition metal trichalcogenides (TMTC), for instance, are interesting when compared with the well-studied dichalcogenides (TMDC’s) because of their quasi-1D properties stemming from a reduced in-plane structural symmetry. This character gives the TMTC class of materials strong anisotropies in their electrical and optical properties [19–21]. This anisotropy lends an additional degree of freedom in the

fabrication of next generation electronics such as high mobility transistors benefiting from reduced backscattering from hot electrons [22] and novel integrated digital inverters [23]. Titanium trisulfide (TiS₃), in particular, has gained recent attention as it presents a robust direct band gap of ∼ 1 eV [24–26] which varies little with layer thickness or stacking order [25].

Table 1.1: Summary of properties of different 2D materials relevant for nanodevices applications.

2D Materials Crystal class Electronic structure Bandgap (eV) Mobility (cm²V⁻¹s⁻¹)

Graphene Hexagonal Semimetal NA 2,000-5,000 (supported)[8]

200,000 (freestanding)[27]

TMDCs (MoS2) Hexagonal Semiconductor From 1.29 for bulk (indirect) < 320 for electrons, < 270 to 1.96 for monolayer (direct)[28] for holes [29]

Phosphorene Orthorhombic Semiconductor From 0.3 for bulk (direct) <10,000-26,000 (monolayer) [30]

to 1.75 for monolayer (direct) 400-4,000 (few-layer) [30]

h-BN Hexagonal insulator 5.971 (direct)[31] NA

Silicene Hexagonal Semiconductor 0.002 [32] 100 [33]

Antimonene Hexagonal Semiconductor 1.6 [16] 800 (value obtained in

this work)

Germanene Hexagonal Semiconductor 0.003 [32]

The absence of an energy bandgap makes graphene less desirable for use in FET’s switching settings that require high on-state currents but low off-state currents. Hence, the field has expanded into other 2D materials, predominantly semiconducting TMDC’s because they possess bandgaps in the range of 1-2 eV, leading to a high degree of electrostatic control [34] and scalability for nanoscale transistors [35], exquisite sensing capabilities [36], high breakdown voltages [37], tunable optical properties [9,38], a high degree of mechanical flexibility [39] and the possibility of engineering new materials through the realization of van der Waals heterostructures [40]. Many of these results indicate that these crystals give promising perspectives to the next-generation nanoelectronics, spintronics and valleytronics. Yet, still this research area encourages unceasingly wide exploration of TMDC’s crytstals’ properties and establishes strong motivation for further studies in this matter.

So far, more than ten different 2D semiconductors (with bandgap values spanning from a few millielectronvolts up to several electronvolts) (see Table1.1) have been experimentally isolated and there are potentially hundreds more that could be isolated in the near future. Because of this broad catalogue of materials, it is always possible to find a 2D semiconductor that is optimal for a certain application. Table1.1shows a comparison of some relevant properties of the most studied 2D materials. Moreover, most of the 2D semiconductor families studied so far have also shown interesting phenomena, some of them observed for the first time in these 2D systems.

1.1. ELECTRICAL CONTACTS TO 2D MATERIALS

These recent progress suggests that such atomically thin 2D materials could be one pathway for electronic devices in the future [12,41,42].

Figure 1.1: Schematic of a typical 2D semiconductor FET. The figure states key issues affecting 2D device performance related to channel materials, metal contact, 2D dielectric interfaces, and short channel effects. (extracted from [43]).

Figure1.1illustrates the key issues observed in a typical 2D semiconductor FET. These issues are arranged into four main categories: quality of channel materials, metal contacts, interfacial effects, and short channel effects. Although many of these issues are observed in traditional bulk semiconductor devices, the severity and impact of these issues on the device performance are expected to be different in the case of 2D semiconductors. For instance, because electrons in a 2D FET are confined in close proximity to the gate dielectric and substrate interfaces, the effects of interfacial scattering are more severe in comparison with traditional FETs made of bulk semiconductors. Moreover, there are some differences in the device structure and fabrication processes between 2D FETs and traditional FETs. For example, in most 2D FETs, the source/drain metal contacts the 2D semiconductors directly. On the other hand, highly doped source/drain regions are formed before metal contacts in traditional FETs. As a result, it is important to understand and solve major problems with 2D FETs in unique ways, so that optimized device performance may be achieved.

1.1 Electrical contacts to 2D materials

Making excellent electrical contacts is a prerequisite to obtain high performance for any of the 2D devices. The quality of the electrical contacts - quantified through contact resistance - is a critical issue for many electronic and optoelectronic devices. Their ultrathin nature and inert smooth surface make it difficult. In short-channel devices, the relative contribution of contact

resistance to the whole device resistance will increase when compared to long-channel devices, and eventually contact may even dominate the performance of the whole device.

Therefore, it is critical to achieve low-resistance contacts to 2D semiconductors. However, it is quite a challenge to achieve good contact for 2D semiconductors for the following reasons. First, the pristine surface of 2D materials is usually intact, without unsaturated atoms, which makes it difficult for them to form strong interface bonds with contact metals, leading to an increased contact resistance. Second, the commonly used approach to decrease contact resistance in silicon and III-V semiconductor electronics is degenerate substitutional doping. This doping technique is difficult to apply to 2D semiconductors because it will inevitably modify the properties of the 2D semiconductor channels, due to the ultrathin 2D-semiconductor body. Third, the contact areas be- tween 2D semiconductors and contact electrodes are usually small, especially for small-dimension devices, which results in large contact resistance. In the past several years, researchers have developed several strategies aimed at improving the contacts in 2D semiconductor electronics (Figure1.2).

Figure 1.2: Current strategies to improve electrical contacts in 2D semiconductor FETs. a) Play with types of contact metals and processing recipe. b) Doping of contact areas. c) 1T-2H phase-engineered in-plane heterojunctions. d) Graphene top contact. e) Graphene edge contact. f) Inserting a tunnel layer between 2D semiconductors and contact electrodes. (extracted from [43]).

Figure1.2(a) shows a typical structure of a back-gated FET using 2D semiconductors as channel materials, which are electrically contacted by two contact electrodes (i.e., source (S) and drain (D)). The second approach to reduce Rc is to dope the contact area, as shown in Figure 1.2(b). Figure1.2(c) shows another approach to achieve low R_c contact in TMDC transistors, i.e., formation of 1T/2H phase junctions. TMDCs have different phases, for example, 1T and 2H, which possess distinct properties. For example, 2H-phase WSe₂ and MoS₂are semiconductors, while metastable 1T-phase WSe₂ and MoS₂ are metals. It has been noted that 1T and 2H phases have reasonably similar structures and lattice constants, and can transform to each other reversibly. Finally, using graphene is another effective approach to achieve good contacts to 2D

1.1. ELECTRICAL CONTACTS TO 2D MATERIALS

semiconductors. There are two kinds of graphene based contacts, i.e., top contact (Figure1.2(d)) and edge contact (Figure1.2(e)).

1.1.1 Lateral heterostructures

Heterojunctions between three-dimensional (3D) semiconductors with different bandgaps are the basis of modern light-emitting diodes [44], diode lasers [45] and high-speed transistors [46].

Creating analogous heterojunctions between different 2D semiconductors would enable band engineering within the 2D plane [6,47–49] and open up new realms in materials science, device physics and engineering.

Lateral heterostructures can also be grown by a variety of methods. Thus, CVD-grown graphene was lithographically patterned and etched away, and h-BN was grown via CVD, forming lateral 1D heterojunctions. Beyond graphene and h-BN, lateral heterostructures based on 2D TMDCs can be disruptive for integrated optoelectronic devices. Although direct growth favors TMDC alloys because of a similar chemistry and a small lattice mismatch between different TMDCs [50], two-step epitaxial growth. The growth of such lateral junctions will allow new device functionalities, such as in-plane transistors and diodes, to be integrated within a single atomically thin layer.

Figure 1.3: Schematic diagrams of a device with a 1T’/2H phase homojunction. (B) AFM image of a device with the 1T’/2H phase homojunction in MoTe2. (C) Raman mapping images of 1T’

(A_g) and 2H(E_2g) vibrational modes in the device channel in (B). (D) Source-drain current I_SD characteristics for gate voltage V_G ranging from -60 V to 60 V. (E) Arrhenius plots of the conductance. (F) Field-effectmobility as a function of temperature (extracted from [51]).

Most of transition metal dichalcogenides possess two typical phases, i.e., semiconducting 2H

and metallic 1T (1T’) phases. An alternative approach that makes use of the intrinsic metallic behavior of a metastable phase of MoS₂, based on phase engineering, to design low-resistance contacts to realize FETs is demonstrated by Kappera et. al. [6] (see Figure1.2(c)). They locally modified by chemical means the arrangement of S atoms in the crystalline structure of a few layers of semiconducting MoS₂, and then deposited metallic contacts on these converted regions.

This decreased the energy barrier between the external circuit and the semiconducting MoS₂ flakes, a marked improvement towards the theoretical minimum contact resistance value of TMDCs (200 - 300 Ωµm at zero gate bias). The low contact resistance is attributed to the atomically sharp interface between the phases and to the fact the work function of the 1T phase and the conduction band energy relative to vacuum level of the 2H phase are similar (∼ 4.2 eV) [6]. However, the wet chemical method involved in the device-fabrication process may limit its efficient applications. According to this issue, Cho et al. [51] developed a laser-irradiation-induced phasetransformation process to fabricate a 1T’/2H MoTe₂ ohmic lateral homojunction contact, as shown in Figure1.3. The 1T’/2H contact increases the carrier mobility over 50 times than the 2H/2H contact.

1.2 van der Waals heterostructures

Once the atomic layers are attained from whichever method best suits the experiment, it is necessary to assemble them into the desired stacking order to create the desired van der Waals heterostructures. Similar to the synthesis of individual layers, both top-down and bottom-up strategies can be applied to heterostructure assembly [7,40]. Semiconducting TMDC’s could be combined with graphene, h-BN, or other 2D materials, and form hybrid all-2D electronics on flexible substrates [40]. These architectures have already been realized by mechanically transferring one monolayer on top of another. However, the artificial fabrication of van der Waals heterostructures also has major limitations; first, there is little control over crystal orientation while stacking on top of each other and second, the sample can suffer damage by unwanted air bubbles or adsorbates between the stacked layers. Moreover, large scale isolation methods lique chemical or liquid phase exfoliations have problems to provide suspensions whith platelets consisting of van der Waals heterostructures with high quality.

An alternative technique, which potentially allows mass production of such structures (i.e., sequential growth of monolayers) comes with its own limitations and is presently in its infancy.

Nevertheless, a large variety of novel experiments and prototypes have already been carried out with van der Waals heterostructures, which indicates that these materials are versatile and practical tools for future experiments and applications.

In this thesis, we present an extensive theoretical and experimental characterizations of franckeite which is a naturally occurring and air stable van der Waals heterostructure. Thin layers of heterostructures based on alternating layers of SnS₂-based and PbS-based 2D layers

1.3. NEW PROMISING 2D MATERIALS

Figure 1.4: Many functional van der Waals heterostructures can be created. What started with mechanically assembled stacks (top) has now evolved to large-scale growth by CVD or physical epitaxy (bottom) (extracted from [52]).

(with remarkably high crystalline and stacking order) by mechanical and liquid phase exfoliation were isolated. As the bulk material is already composed of these alternating SnS₂ and PbS layers, the exfoliation process minimizes stacking missorientation and avoids interlayer adsorbates in the isolated nanosheets of franckeite [53]. Hence, franckeite can be considered as a naturally occurring vdW heterostructure analog of its synthetic cousin.

1.3 New promising 2D materials

1.3.1 Germanene

Graphene’s success has shown not only that it is possible to create stable, single-atom-thick sheets from a crystalline solid but that these materials have fundamentally different properties than the parent material. Other group IV layered lattices may maintain appreciable conductivity when the atoms are in the sp³ hybridized state. Recently, single-layer-thick sp² and sp³group IV systems have attracted considerable theoretical and experimental interest [54,55].

The most obvious alternatives for graphene are the group IV elements, i.e. silicon and germanium [56]. The electron configurations of germanium, silicon and carbon are very similar since all three elements have four electrons in their outermost s and p orbitals. The energetically most favourable crystal structure of silicon and germanium is the diamond structure [57]. The diamond lattice consists of two interpenetrating f cc sub-lattices and each atom of these f cc sub-lattices is surrounded by four neighbours. The covalent bonds between the atoms are all

equivalent and have a hybridized s, p_x, p_y, p_z character (sp³). The first reports on the synthesis of silicene date back to 2010 [58,59], followed by germanene in 2014 [15,32,55].

Free standing germanene does not exist, but it is predicted to be a buckled honeycomb 2D Dirac material [17] with extremely high mobilities of its charge carriers [60]. The spin-orbit coupling along with the ∼ 0.64 Å buckling opens up a ∼ 24 meV band gap at the Dirac points significantly higher than in silicene (1.55 meV); this, together with the non-trivial topological properties, might result in a quantum spin Hall effect detectable at room temperature [61].

Typically, since germanium is currently used as a performance booster in thin FET channels, perspectives of using germanene for scaling down beyond the 5 nm node while increasing the speed and lowering the energy consumption of electronic devices appear very promising.

1.3.2 Antimonene

The 2D materials of group V elements are gaining considerable interest because of the unique properties. In the last two years phosphorene has generated a considerable attention due to the fact that, as a function of the number of layers, it features a direct optical band gap in the range 0.5-1.8 eV, suitable for THz optoelectronic applications [10,62]. However, its ambient stability represents a major drawback for the development of the real applications [63]. In contrast, a member of the group V in the periodic table, antimonene, i.e. a single layer of antimony, presents an outstanding stability under ambient conditions [16].

In a recent study, Ares et. al. reported both micromechanical exfoliation of antimony down to the single-layer regime and experimental evidence of its stability. They demonstrated that single/few-layer antimony flakes are highly stable in ambient conditions showing mechanical stability upon origami nanomanipulation and no degradation over month periods. Density functional theory (DFT) simulations mimicking ambient conditions confirm the geometrical experimental findings and predict a bandgap of 1.2-1.3 eV within the range of optoelectronics applications [16]. They have used optical microscopy used to study the optical properties of few- layer antimonene flakes and quantitatively estimate their thickness in a fast and nondestructive way. Antimonene has been recently isolated by the liquid phase exfoliation [64], that has been successfully applied to generate single- or few-layer (FL) samples of several 2D materials on large scale [65].

These experimental works were preceded by a considerable number of theoretical calculations that predicted extraordinary physical properties for antimonene such as high carrier mobility, thermal conductivity, and strain induced band transition, among others [66–68]. Moreover, the most stable antimonene holds buckled honeycomb structure with much stronger spin-orbital coupling (SOC), which brings exotic fundamental properties for photonics and spintronics. Up to now, theoretical works on antimonene largely exceeds experimental works.

Theoretical works on antimonene have been subdivided into two fronts, quite clearly separated in time (and in researcher communities), and can be found in the literature. The most recent

1.3. NEW PROMISING 2D MATERIALS

works (theoretical and experimental) refer to monolayer antimonene (or occasionally bilayer antimonene) and can be found in the context of new 2D crystals [16,66]. The second one, which goes a few years back in time, refers to few-layer (FL) antimonene (or Sb thin films), and can be found in the context of 3D topological insulators [69]. The physical properties of antimonene change quite drastically from the monolayer to FL systems (these generally considered to be in the range of 10-100 layers) and both deserve a separate discussion.

C

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2

T

HEORETICAL

B

ACKGROUND

2.1 Introduction

Q

uantum mechanical modeling has become an invaluable tool in materials research being, nowadays, the most accurate computational method used for solving emerging problems in materials science, clean energy, nanotechnology, catalysis, etc. Such simulations provide quantitative predictions and also describe the general trends of physical and chemical properties of a broad range of systems. By solving the quantum mechanical equations of a system, we can achieve a greater understanding of a large number of physical phenomena related to the properties of real material systems.

The properties of a quantum mechanical many-body system are calculated from the many- particle Schrödinger equation: ˆH|α〉 = Eα|α〉, with ˆH = ˆT + ˆV , where ˆT and ˆV are the kinetic energy operator and the potential energy operator for all the particles of the system, respectively.

This equation contains the information about the properties of the physical system in question.

The degrees of freedom of this problem are the positions and momentums of all the particles.

Considering a quantum physical system with Z electrons per atom and N atoms, the position of the particles is describe by Z · 3N variables. Unfortunately, the many-body nature of the Schrödinger equation makes itself extremely difficult to solve and for large N the solution of it becomes impossible. Hence, to avoid insolubility of the Schrödinger equation is necessary some approximations to render the problem soluble (more details in sec.A.1).

In this sense, the density functional theory (DFT) emerged as a powerful tool for electronic structure calculations [70]. DFT reduces (see sec.A.1) the problem of Z · 3N variable to one with only 3. This theory is the base of the ab initio calculations and, in principle, provide information only on the ground state of the system. DFT implementations generally exhibit a good balance

between accuracy and computational cost. Despite the intensive software development, the computational cost of ab-initio methods still remains very expensive. This originates from the complexity of electrons represented by wave functions, hence possessing an inner structure with widely variable properties compared to simple particles used in classical molecular dynamics.

The requirement of self-consistency is only one consequence of the quantum nature, which slows down such methodology. Although many concepts exist to weaken this impact, there is a practical limitation to sizes of systems, at present 1-2 nm, treatable ab-initio. This, however, is not the length scale on which one discusses functional materials.

If one is interested in the electronic structure of large systems or if interatomic potentials are not accurate enough for the desired application, one option is to turn to semiempirical methods, which lie conceptually between empirical force fields and first principles methods and allow for the treatment of tens of thousands of atoms. Semiempirical electronic structure methods can be understood as direct approximations to more accurate methods (usually DFT or Hartree- Fock), but additionally include empirical parameters that can be tuned to reproduce reference data (experimental or theoretical). One particular example of a semiempirical method is the tight-binding (TB) electronic structure method.

The tight-binding model is a simple and computationally feasible single-particle model that provides a reasonable description of occupied states in any type of crystal (metals, semiconductors, and insulators) and often also of the lowest lying conduction states [71]. The Hamiltonian matrix is then readily constructed and diagonalized. Tight-binding method is specially relevant in the study of nanoscopic systems, for instance graphene [72, 73], graphene-related materials and nanostructures [74–76] and the transition metal dichalcogenides (TMD) [77–79].

The tight-binding method is an efficient formalism used to create Hamiltonians in localized basis. This Hamiltonian can be used to compute ballistic quantum transport within the coherent transport approximation i.e no inelastic events. Landauer formalism and the partitioning scheme require a localized representation of the electronic structure. We have seen that tight-binding is very efficient dealing with big systems. Therefore, with both methodologies it is posible to explore the electronic transport in a great number of systems. In this thesis, we have compute the tight-binding quantum transport, aided by DFT calculations, of the lateral heterostructure formed by two most commom crystallographic phases of MoS₂ monolayer crystals: the stable semiconducting 2H phase and the metastable metallic 1T phase. All the transport calculations have been done using the code ANT.1D [80].

Other theoretical method used in this thesis is the technique called "Monte Carlo Simulation"

or kinetic Monte Carlo (KMC). KMC algorithms are powerful techniques to study the dynamics of a system of particles when the different events that those particles can perform are known as well as their probabilities [81]. The KMC simulations were employed together with DFT calculations to study the thermal stability and the vacancy generation mechanism of the single layer TiS₃ with differents oxygen concentrations on the surface.

2.2. TIGHT-BINDING MODEL

2.2 Tight-binding model

The goal of this section is introduce the basic concepts of the tight-binding model and the terminology and definitions used in this work. The TB model assumes that the electrons are tightly bound to atoms, such that the confining atomic potentials in a crystal are strong. Then, the atomic orbitals of isolated atoms can be used as the single-particle basis, and the tight-binding states are written as linear combinations of them. Moreover, the atomic orbitals of nearby atoms are coupled by small overlaps and tunneling matrix elements that are the relevant parameters of the model.

The starting point for all semi-empirical approaches is the physics. In metals, for example, the electrons are almost free and so we can treat the single particle states in terms of plane waves. We could also take a very different approach and assume the states in a crystal look like combinations of the wavefunctions of isolated atoms. We might imagine this is more likely to be the case in insulators or semiconductors.

2.2.1 Electrons is solids

The motion of electrons moving in a solid is governed by:

H = ˆˆ T_e+ ˆV_ee+ ˆV_en. (2.1) A mean-field approach simplifies the system mapping the N-electron equation to a set of single- electron equations for the motion of an electron in the mean field of the nuclei and the other N − 1 electrons. We represent the electron-electron and the electron-nuclei interactions with an effective one-electron potential V (r) which has the same periodicity of the lattice

V (r) = V (r + R), (2.2)

for all Bravais lattice vectors R. In this way, we have to find solutions of a Schrödinger equation of the form

Here, we will solve the single particle Schrödinger equation for the states in a crystal by expanding the Bloch states in terms of a linear combination of atomic orbitals.

2.3 Linear Combination of atomic orbital (LCAO)

2.3.1 Crystal and atomic hamiltonians

In a crystal, we take the single particle hamiltonian to be

H = Hat+∆U (2.3)

where H_atis the hamiltonian for a single atom and∆U encodes all the differences between the true potential in the crystal and the potential of an isolated atom. We assume∆U → 0 at the center of each atom in the crystal.